#835

Medium

LeetCode Problem Workspace

Image Overlap

Image Overlap requires calculating the overlap between two binary matrices by translating one image over the other.

Problem Statement

You are given two binary square matrices, img1 and img2, both of size n x n. The matrices consist of only 0's and 1's. You can translate img1 by sliding it in any direction: left, right, up, or down. After each translation, calculate how many positions have a 1 in both img1 and img2. If img1 is translated outside the matrix borders, the 1's are erased. Your task is to determine the maximum overlap that can be achieved by any translation.

The translation of img1 does not involve rotation, and any 1 bits that are moved outside the boundaries are ignored. The goal is to compute the highest overlap of 1's between the two matrices after applying all possible translations. The matrices' sizes range from 1x1 to 30x30, with each element being either 0 or 1.

Examples

Example 1

Input: img1 = [[1,1,0],[0,1,0],[0,1,0]], img2 = [[0,0,0],[0,1,1],[0,0,1]]

Output: 3

We translate img1 to right by 1 unit and down by 1 unit.

The number of positions that have a 1 in both images is 3 (shown in red).

Example 2

Input: img1 = [[1]], img2 = [[1]]

Output: 1

Example details omitted.

Example 3

Input: img1 = [[0]], img2 = [[0]]

Output: 0

Example details omitted.

Constraints

n == img1.length == img1[i].length
n == img2.length == img2[i].length
1 <= n <= 30
img1[i][j] is either 0 or 1.
img2[i][j] is either 0 or 1.

Solution Approach

Brute Force Approach

One way to solve this problem is by iterating over all possible translations of img1 over img2. For each possible translation, count the number of overlapping 1's and keep track of the maximum overlap encountered. This approach checks each possible shift, both horizontally and vertically, and calculates overlaps, making it simple but inefficient.

Optimized Approach with Sliding Window

An optimized approach involves reducing the number of shifts by using a sliding window. This method only checks for the shifts that are likely to result in maximum overlap based on the positions of 1's in the matrices. This allows the solution to avoid unnecessary checks and improve performance by focusing on valid translations.

Matrix Comparison Using Hashing

Another optimization technique involves representing the overlap check as a hashing problem. By converting parts of the matrices into hashable keys, we can quickly determine the overlap between different parts of img1 and img2. This method speeds up the comparison process by reducing it to simple hash lookups instead of iterating through matrix entries directly.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The brute force approach has a time complexity of O(n^4) due to the need to check all possible shifts for an n x n matrix. The optimized sliding window approach reduces the complexity to O(n^2), while the matrix comparison using hashing can potentially lower this further depending on the efficiency of the hashing function used. Space complexity varies from O(1) to O(n^2), depending on the approach chosen for storing and comparing matrix data.

What Interviewers Usually Probe

Ability to recognize and optimize brute force solutions.
Experience with sliding window or hashing techniques for optimization.
Understanding of matrix manipulation and translation operations.

Common Pitfalls or Variants

Common pitfalls

Failing to account for translations outside the matrix boundaries.
Overlooking the need to optimize the brute force approach for larger matrices.
Misunderstanding the problem as one requiring rotation, when only translation is allowed.

Follow-up variants

Handling larger matrices where brute force is inefficient.
Including additional constraints such as specific translations or partial overlaps.
Adapting the problem for non-square matrices or non-binary values.

FAQ

What is the best approach to solve Image Overlap?

The best approach is an optimized one, such as using a sliding window or hashing, to reduce the computational cost from a brute force O(n^4) to O(n^2).

How does the Image Overlap problem relate to sliding window techniques?

In Image Overlap, sliding window techniques help optimize the solution by focusing only on potential overlapping positions, avoiding unnecessary checks.

Can the matrices in Image Overlap problem be rotated?

No, the problem specifically restricts translations to shifts without rotation. This is a key distinction in how the problem is solved.

What are common mistakes in solving the Image Overlap problem?

Common mistakes include failing to optimize the brute force solution, misunderstanding the translation rules, and not correctly handling matrix boundaries.

How does GhostInterview prepare you for Image Overlap?

GhostInterview guides you through solving Image Overlap by helping you choose efficient techniques like sliding windows and hashing, ensuring you avoid common pitfalls.

terminal

Solution

Solution 1: Enumeration

We can enumerate each position of $1$ in $\textit{img1}$ and $\textit{img2}$, denoted as $(i, j)$ and $(h, k)$ respectively. Then we calculate the offset $(i - h, j - k)$, denoted as $(dx, dy)$, and use a hash table $\textit{cnt}$ to record the number of occurrences of each offset. Finally, we traverse the hash table $\textit{cnt}$ to find the offset that appears the most, which is the answer.

class Solution:
    def largestOverlap(self, img1: List[List[int]], img2: List[List[int]]) -> int:
        n = len(img1)
        cnt = Counter()
        for i in range(n):
            for j in range(n):
                if img1[i][j]:
                    for h in range(n):
                        for k in range(n):
                            if img2[h][k]:
                                cnt[(i - h, j - k)] += 1
        return max(cnt.values()) if cnt else 0

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